APPENDIX E

Transport Properties

The diffusion velocities, viscous stresses, and heat fluxes appearing in the conservation equations of fluid dynamics [see Appendix C or D], which are determined by the molecular transport of mass, momentum, and energy, respectively, must be calculable in order to render the conservation equations soluble. The fact that these quantities cannot generally be directly related to the other variables appearing in the conservation equations is readily apparent, since the transport properties involve higher velocity moments of the distribution function [see, for example, equation (D-28)]. For near- equilibrium systems, Enskog has employed a series expansion of the velocity- distribution function about the Maxwellian distribution in order to obtain from the Boltzmann equation explicit expressions for the transport vectors (and tensor) in terms of the gradients of the dependent variables of fluid dynamics. The resulting closed set of equations constitute the Navier- Stokes equations, which have been found to be applicable for considerably large deviations from equilibrium.* Since Enskog’s rigorous derivation of the Navier-Stokes equations necessarily is very long, a physical discussion somewhat analogous to those in [1]-[3] is presented here, and the reader is referred to [ 3 ] - [ S ] for the exact treatment. In view of the fact that simplified

* Higher-order approximations have not been proven to be useful; when the Navier- Stokes equations are invalid, the most satisfactory procedure is to solve the Boltzmann equation by methods that do not rely on the assumption of near-equilibrium flow.

628

E.l . Collision Integrals 629

analyses, while possibly giving more insight into the problem, lead to in- accurate expressions for the transport coefficients (the constants of propor- tionality between the fluxes and the gradients), the results of the exact theory also will be quoted here. Reference [6] may be recommended as a thorough textbook on transport phenomena.

In the following section molecular collisions are discussed briefly in order to define the notation appearing in the exact expressions for the transport coefficients. Diffusion is treated separately from the other trans- port properties in Section E.2 because it has been found [7] that closer agree- ment with the exact theory is obtained by utilizing a different viewpoint in this case. Next, a general mean-free-path description of molecular transport is presented, which is specialized to the cases of viscosity and heat conduction in Sections E.4 and E.5. Finally, dimensionless ratios of transport coeffi- cients, often appearing in combustion problems, are defined and discussed. The notation throughout this appendix is the same as that in Appendix D.

E.1. COLLISION INTEGRALS

The exact kinetic theory of dilute gases leads to expressions for transport coefficients in terms of certain quantities called collision integrals, which depend on the dynamics of binary intermolecular collisions.* These integrals are defined in this section.

Since the collisions of predominant importance in transport processes are elastic and do not involve chemical reactions, a potential cp may be defined, the negative gradient of which is the force between the two inter- acting molecules. The interaction may accurately be treated classically for all molecules at room temperature and above. Consideration is restricted to central forces, for which cp depends only on r, the distance between the mass centers of the molecules [q = q ( r ) ] . Useful results for more general potentials (which, rigorously, are required to describe interactions of polar molecules) have not been obtained. The arbitrary constant in the potential is defined by cp(.o) = 0.

It is easily seen from classical mechanicst that the binary collision problem is mathematically equivalent to a one-body problem in which a body with the reduced mass

p = p i i = mimj/(mi + mi) (1)

(where mi and mj are the masses of the colliding molecules) moves in the

* In dilute gases (those obeying the ideal gas law), ternary collisions occur so seldom that they are of no consequence in transport.

See, for example, Chapter 3 of 181 or Chapter 1 of [5].

630

\ \ Transport Properties

FIGURE E.1. Schematic diagram of the equivalent one-body collision.

fixed potential q ( r ) . A schematic diagram of the collision in this representa- tion is shown in Figure E.l. Here r , is the distance of closest approach of the molecules, x is the deflection angle in the collision, b is the impact parameter (the distance of closest approach if there were no potential), and g is the rela- tive approach velocity of the two molecules when they are infinitely far apart. From an analysis of the collision [ I S ] , [ S ] , it is found that ~ ( b , g) is given by

= 7r - 2”; (;) [I - (A) - ( ; ) ] - 1 ‘ 2 d r > (2)

where r , is determined by the solution of the equation

d r , ) = (PLS2/2” - (b2/r31. The collision integrals are defined as

(3)

where the dimensionless velocity # is

From equations (1)-(5) it is apparent that in addition to the powers k and 1, the ni,”.” depend on mi, m j , T and the form of the potential q ( r ) . Therefore, only q ( r ) remains to be found in order to determine completely the collision integrals.

Theoretical calculations of q ( r ) from molecular structure are difficult and necessarily quite approximate in nature. It has been found to be more useful to assume a reasonable functional form for q ( r ) with adjustable constants and to choose these constants to fit experimental data. Experiments yielding information about q ( r ) are (1) measurements of transport coeffi- cients [giving q ( r ) at low energies (large r ) ] and (2) cross-section measure- ments by molecular-beam scattering [giving q ( r ) at high energies (small r ) ] . A number of different functional forms q ( r ) have been employed, most of which are described in [ S ] .

E.2. Diffusion 63 1

The Lennard-Jones potential [4], [S], [9] (sixth-power attraction, twelfth-power repulsion) is quite realistic and appears to be the one most commonly used in practice. This potential contains two adjustable param- eters (a “size” and a “strength”), which are defined and listed for various chemical compounds in [S], [6], and [9]. The collision integrals appearing in the first approximations to the transport properties are tabulated as functions of useful dimensionless forms of these two parameters in [S], [6], and [9]. Similar tabulations for other potentials may also be found in [S].

E.2. DIFFUSION

E.2.1. Physical derivation of the multicomponent diffusion

An equation determining the diffusion velocities Ti for gas mixtures may be derived by the following physical arguments. Since the total momen- tum is conserved in collisions, the momentum of molecules of type i can be changed only by collisions with molecules of other types ( j # i) and by body forces acting on species i. It is to be expected that on the average, the momen- tum transferred to an i molecule in a collision between molecules i and j is approximately pij(oj - Ti) because (Vj - Ti) is the average relative velocity of molecules of species i and j . The fact that the average momentum trans- ferred from i to j must equal the negative of that transferred from j to i implies that the average momentum transferred is symmetrical in i and j , whence the reduced mass pij enters the expression. The body force acting on molecules of type i in a unit volume element is clearly p q f i . Hence, the net rate of change of momentum of molecules of type i per unit volume is

equation 1101

N

Ti) + p q f i , i = 1,. . . j = 1

where vi j is the total number of collisions per unit volume per second be- tween molecules of kinds i and j .

This change in momentum of species i is manifest in changes in both the random velocity and the order velocity of the molecules. The rate of change of the ordered momentum is pY,Dv,/Dt, where the substantial derivative with respect to time is

DIDt = d / d t + vo - V,.

Since (neglecting the off-diagonal elements of the pressure tensor*) the partial

* These assumptions, which are usually tacit in elementary treatments, are required in the order of approximation used here, as may be seen, for example, from Section 7.3,468-469 of [ S ] .

632 Transport Properties

pressure of species i, pi is physically the momentum of molecules of type i transported per second across a surface of unit area traveling with the mass- average velocity of the fluid [see, for example, equations (D-15) and (D-20)], it follows that V,p, is the rate of change of momentum of random motion of molecules of type i per unit volume. Therefore, the quantitity Ti also is given by the relation

Ti = pqDv,/Dt + V,p,, i = 1 , ..., N. (7) Under the assumption that the translational temperatures of all species, are the same,* equations (D-9), (D-20), and (D-22) show that Dalton’s law of partial pressures is valid, that is,

p i = X , p , i = l , . . . , N, (8)

whence

V,p, = pV,Xi + X,V,p, i = 1 , . .. , N. (9) When the off-diagonal elements of the pressure tensor are neglected as above,* the momentum equation, equation (D-35), becomes

N

pDv,/Dt = -V,p + C Tfj. (10) ;= 1

Substituting equations (6), (9), and (10) into equation (7) yields

where use has been made of the identity = 1. Equation (11) is essen- tially the desired diffusion equation.

The product pijv i j in equation (1 1) may be related to the binary diffu- sion coefficients by considering the limiting case of a constant-pressure process in a two-component system with no body forces, for which equation (1 1) reduces to

V J l = (PlZVIZ/P)(~2 - TI), (12) where the subscripts 1 and 2 identify the two components. The diffusion velocity Vl of species 1 with respect to the number-weighted average velocity of the mixture is

(vg + V1) - [X,(V, + V1) + Xz(vg + V,)] = XZ(V1 - Vz), (13) V;

* See previous footnote

E.2. Diffusion 633

in which the last equality follows from the identity (1 - X,) = X 2 . Sub- stituting equation (13) into equation (12) leads to the result

V,Xl = -(Pl2vl2/x2P)~;. (14)

Dl2V,X1 = -X,P; (15)

The binary diffusion coefficient D,, for species 1 and 2 is often defined by the equation

under the present conditions; that is, it is the constant of proportionality between the mole-fraction gradient and the number flux with respect to the number-weighted average velocity. A comparison of equations (14) and (1 5) then shows that

which implies that, in general, D12 = XlX2P/Pl2V12~

D. . = XiX.p/p. . \ l . . J I J I J '

Utilizing equation (16) in equation ( 1 1) yields

N

+ - C yq(f i - fj), i = 1 , . . ., N,

as the final form of the multicomponent diffusion equation. Equation (17) states that concentration gradients may be supported by diffusion velocities, pressure gradients (when the mass fractions differ from the mole fractions) and differences in the body force per unit mass on molecules of different species. It can be shown from the rigorous kinetic theory that equation (17) is valid to first order in a Sonine polynomial expansion off, when thermal diffusion (diffusion resulting from temperature gradients) is negligible [S]. The multicomponent diffusion equation obtained from the complete kinetic theory is [5]

i = 1 , . . . , N , (18) where DT, is the thermal diffusion coefficient of species i in the multicompon- ent mixture. The physical viewpoint presented here is easily extended to show how thermal diffusion (the Soret effect) may arise, and it even enables one to determine the signs of the thermal diffusion coefficients [7]. In quoting equation (18) in Chapter 1, the bars over the diffusion velocities and the subscript x in V, are omitted.

634 Transport Properties

E.2.2. Simplified diffusion equations

For binary mixtures it may be shown that equation (18) reduces to

Y l o l = - D 1 2 [ V,Yl - (;:) ~ (Yl - X , ) (y)

From equation (19), we see that the familiar Fick’s law of diffusion

Ylvl = -Dl ,V ,Yl ,

is rigorously valid when (1) the mixture is a binary mixture, ( 2 ) thermal diffusion is negligible, ( 3 ) the body force per unit mass is the same on each species (f, = f,), and (4) either pressure is constant (V,p = 0) or the molec- ular weights of both species are the same (Y, = X Because of the complex forms of equations (18) and (19), in many of the applications discussed in this book equation (20) is utilized in order to make the governing equations tractable. The use of equation (20) in combustion problems is partially justified by the facts that conditions 2 and 3 are almost always valid approxi- mations and V, p / p usually is sufficiently small that conditions 4 is satisfied. Requirement 1 may be replaced by somewhat less stringent conditions (for example, that the binary diffusion coefficients of all pairs of species in the multicomponent mixture are equal), but, of the four conditions listed above, 1 (or its equivalent) usually is the most difficult to justify in combustion problems.

If the condition 1 is not imposed, then with approximations 2, 3, and 4, equation (18) reduces to

which has been called the Stefan-Maxwell equation [6]. Equation (21) is more complicated to employ than equation (20), mainly because, in general, it is not readily soluble for diffusion velocities in terms of concentration gradients; it provides concentration gradients in terms of diffusion velocities. However, there are some combustion problems for which use of equation (21) will not introduce excessive complexity in seeking analytical solutions. Moreover, when electronic computers are employed to calculate numerical solutions to the differential equations of combustion, use of equation (21) should not be appreciably more complicated than use of equations involving multicomponent diffusion coefficients (see Section E.2.4) particularly in one-dimensional problems. Nevertheless, such calculations seldom have been based on equation (21).

E.2. Diffusion 635

E.2.3. Binary diffusion coefficients

Equation (16) constitutes a physically derived expression for the binary diffusion coefficients Dij. This equation may be written in a more useful form by expressing the number of i-j collisions per unit volume per second (vij) in terms of more basic molecular parameters. Since there are ni molecules of type i per unit volume, vij = ni/Tij, where T~~ is the average time between collisions of a given molecule of kind i with molecules of kind j . Letting oij denote the collision cross section for molecules of types i and j [for example, oij = n(ri + rj)’ for hard spheres with radii ri and r j ] and fi i j denote the average relative velocity of molecules of kinds i and j, we may consider the given molecule of kind i to sweep out a volume oij fi ij per second, such that the given i molecule will collide with any j molecule in this volume. Since, on the average the i molecule will collide with a j molecule after it has swept out a volume I / F I ~ , where n j is the number of molecules of type j per unit volume, it is clear that the number of collisions of the given i molecule with j molecules per second is l /z i j = nj(oij f i i j) . Hence

(22) v . . I J = n.n.o. . f i . . 1 J 1J 15 = x i X . n 2 0 . . f i . . J I J I J ’

where use has been made of the definition of X i . Substituting equation (22) and the ideal gas law, equation (D-22), into equation (16) yields

D i j = kOT/np..o..fi . . I J IJ LJ (23)

as the equation for the diffusion coefficient. The conventionally quoted result of the exact kinetic theory [ S ] , [9]

is that obtained from a first-order expansion in Sonine polynomials, namely,

D i j = 3k0T/16npi jQ! j31) , (24)

where the collision integral Qij. ’ ) is defined in equation (4). It may be shown [ S ] that Qi; , ’) = aijijij/4 for hard-sphere molecules, in which case it follows that equation (23) is in error only by the constant factor of i. Since it is easily shown from the Maxwellian velocity distribution that

,-- fiij = 2 /8k0T /np i j ,

it follows from equation (23) that (since n - PIT), the pressure and tempera- ture dependence of D i j is D i j - T3’2/p. For realistic intermolecular force potentials, equation (24) gives D i j - T“/p, where, roughly, 1 I c( 5 2. Values of D i j typically lie between cm’/s and 10 cm’/s at atmospheric pressure. Equations (23) and (24) both predict that D i j is independent of the relative concentrations X i and X j and that Dij increases as the molecular weight decreases ( D i j - l/J&). The product

p D i j = 3Wk0T/16pijQ{~,”,

where V E CYz1 X j Wj is the average molecular weight of the mixture

636 Transport Properties

[see equations (D-5)-(D-9)], often appears in combustion problems and may be called the mass diffusivity. References [ S ] , [6], and [9] and references quoted therein may be consulted for tabulations of binary diffusion coeffi- cients for gases. Sources of newer experimental information are quoted in [11] and data are updated continually in a useful periodical [l2]. Binary diffusion coefficients for liquids and solids are not given by equation (24); the reader may consult [6], [13], and [14] for information on the former and [lS] for information on the latter. Additional sources of data on these and other transport properties appear in [16]-[21].

E.2.4. Multicomponent diffusion coefficients

Diffusion equations often are written differently from those given in Section E.2.1 [6]. In particular, multicomponent diffusion coefficients differing from D i j often are introduced so that diffusion velocities may be expressed directly as linear combinations of gradients. The multicomponent diffusion coefficients are defined so that they reduce to D i j for binary mixtures [6]. Use of diffusion equations involving multicomponent diffusion coeffi- cients is being made increasingly frequently.

For liquids and solids, the molecular theory of diffusion is less highly developed than for gases, and, therefore, fundamental reasons for preferring one correct form of the diffusion equation over another do not exist for con- densed phases. However, the theory of diffusion for gases rests on firmer grounds, which enable better assessments to be made of advantages of diff- ering formulations. In particular, it is known that correct statements of multicomponent diffusion equations for gas mixtures involve multicompon- ent diffusion coefficients that depend on mole fractions in a complex manner and that are not necessarily symmetric in the indices i and j [6]. In contrast, the D i j in equation (24) are symmetric in i and j , and the product nDij is independent of mole fractions. This affords greater economy in tabulation for binary diffusion coefficients and also may enable greater accuracy in prediction to be achieved if use is made of equation (18) or (21). Therefore, there exist fundamental reasons for preferring not to introduce multicompon- ent diffusion coefficients for gas mixtures. Notwithstanding this advantage of the formulation in equation (18), there may exist problems whose solutions are facilitated by use of different forms of multicomponent diffusion equa- tions, even if the greater complication in functional dependences of diffusion coefficients is retained. At the current state of development of computational capabilities, problems are not solved with the full equations including precise variations of coefficients; therefore, with few exceptions [22], tests of relative efficiencies of differing precise formulations are unavailable.

A common approximation that has been employed is to introduce an effective diffusion coefficient Di for each species i in the mixture, such that either

(25) -

q V i = - D i V , Y , i = 1, . . . , N ,

E.2. Diffusion 637

or XiPf = -DiVxXi, i = 1, . . . , N [compare equations (15) and (20)]. It is important to realize that if an approximation of this kind is employed, then only N - 1 of the quantities Di can be specified independently; the identities cFYl yoi = 0 and cF=l = 1 may be used in equation (25) to show that Di = cj+i DjVx ?/c,,, V, y. Thus one species must be treated differently from the others if equation (25) is to be applied consistently. A situation in which equation (25) can be derived logically is that in which all species except one, say i = N , are present only in trace amounts, so that 1 - X , G 1. In this case equation (25)is to be used only for i = 1, . . . , N - 1, and it is found that Di = D i N , the binary diffusion coefficient for diffusion through the species present in excess. In more general situations, the formula

(26)

is qualitatively correct but, as emphasized above, can be used properly at best for only N - 1 species. The diffusion velocity of the component for which equation (25) is not used must be obtained from the identity

- N c yvi = 0. i = 1

In general, if equation (25) is introduced for all i, then the diffusion coefficients Di must depend on the diffusion velocities, that is, on the solution to the problem, and the formulation then encounters complexities that prevent it from being very useful. Calculations have been performed in which equations (25) and (26) are employed for all species and the fundamental requirement cf=l y V i = 0 is violated; the results obtained may not be too bad if the degree of violation is not too great.

E.2.5. Thermal diffusion coefficients

The thermal diffusion coefficients D T , appearing in equations (18) and (19) in general may be positive or negative and depend on pressure, temperature, and concentrations. It is almost always found that the dimen- sionless ratio D T , i / ( ~ D i j ) is less than& for all pairs of gaseous species i and j, which implies that thermal diffusion usually is negligible in comparison with the ordinary concentration-gradient diffusion. The complicated expressions for the coefficients of thermal diffusion will therefore be omitted here; the reader is referred to [ IS] and [9] for theoretical results and for useful empirical formulas.

Thermal diffusion coefficients should not be confused with the thermal diffusivity, a quantity defined in terms of the thermal conductivity and re- ferring to conduction of heat (see Section E.5). Thermal diffusion one of the cross-transport effects, is a physical process entirely separate from heat conduction. It tends to draw light molecules to hot regions and to drive heavy molecules to cold regions of the gas. Hydrogen is a species that is

638 Transport Properties

likely to be relatively strongly influenced by thermal diffusion. Fine particles with sizes small compared to the mean free path (see Section E.3) behave like large molecules and migrate toward cold regions under the influence of thermal diffusion; for such particles, as well as for larger particles, this effect often is called thermophoresis. Combustion problems exist, involving hydro- gen or fine particles, in which t.hermal diffusion may be of importance. The phenomenon also may be of significance concerning certain flame instabilities [23]. Numerical solutions of differential equations of combustion by electronic computers occasionally have been obtained with thermal diffusion included. Analytical methods in which the effect is taken into ac- count only in regions of high temperature gradients appear feasible for some combustion problems but currently are unexplored. References to literature or alternative ways to formulate equation (18) with thermal diffusion in- cluded may be found in [6]. Quantities related to D T , i and sometimes used in place of it include the thermal dflision ratio, the thermal dlfSusion factor, and the Sorer coeficient (see [6] for the definitions). In condensed phases, D T , i / ( p D i j ) tends to be larger than in gases [6].

E.3. UNIFIED ELEMENTARY TREATMENT OF TRANSPORT PROCESSES

The rough mean-free path development to be given in this section will be used subsequently for viscous stresses and for thermal conduction ; it may also be applied to diffusion, but the.results in this case are not so satisfactory as are those of the preceding section.

The essential features of the results, including all those needed for the subsequent applications, may be obtained by limiting our attention to a one-component gas. The symbol Q will denote any property of a molecule that can be changed by collisions (Q will later be set equal to momentum and energy), and Q will denote the average value of Q for all molecules. For simplicity we shall consider a geometrical configuration in which @/ax > 0, aQ/ay = aQ/az = 0, whence molecular motion will tend to transport Q in the negative x direction. Let F denote the flux of Q in the fx direction, that is, the net amount of Q per unit area per second transported across a surface normal to the x axis by molecular motion. Our aim will be to compute F , in terms of (dQ/dx),, where thesubscript 0 identifies conditions at x = 0.

It will be assumed that at each collision, both colliding molecules exchange their property Q, equalizing it between them,* and that their resulting Q equals the Q appropriate to the level at which they collide. Both of these assumptions are, at best, approximations that may be valid on the

* “Persistence of velocity” corrections constitute an example of methods of accounting for deviations from this approximation.

E.3. Unified Elementary Treatment of Transport Processes 639

average. As a consequence of these assumptions, F , is determined by the location of the last collision of a molecule before it crosses the plane x = 0. This location depends on the average distance that a molecule travels be- tween collisions, which is called the mean free path 1.

An estimate of the mean free path 1 may easily by obtained by a slight extension of the reasoning given at the beginning of Section E.2.3. Letting j = i (where i will now identify the only species present in the one-component system), we find that the average time between collisions of a given i molecule with another i molecule is (see Section E.2.3) zii = l/(nioiifiii). If the average velocity of i molecules is U i , then clearly 1 = ziiUi = (l/nioii)(Ui/fiii). A rough estimate of the - last factor in this expression may be obtained by assuming that Ci/Uii x ( u ~ / v ~ ) ' ! ' and by adopting a statistically independent pair- velocity distribution (that is, by considering near-equilibrium conditions). The last condition implies that 3 = 2 7 , whence Ui/Ui i % 1/$ and

I = I /$ino, (27) where n = ni and the subscripts on CT have been omitted in the one-component mixture.

It may be expected that, on the average, molecules crossing the plane x = 0 from below will have Q = Q(-1) (that is a value of Q appropriate to the position x = - I ) , and those crossing from above will have Q = Q( + 1). Hence

(28) F , = J - Q ( - l ) - J + Q ( + ! ) , where J - is the number of molecules per unit area per second crossing the plane x = 0 from below and J , is the number per unit area per second cross- ing from above. Assuming that the system is sufficiently near equilibrium that the velocity distribution may be taken to be isotropic, we find that J , = J - = J , where J = nUx, in which f ix is the x component of velocity averaged only over those molecules traveling in the +x direction. It is easily proven by integrating the (isotropic) velocity distribution over a hemisphere that in terms of the average velocity of a molecule Ui E U, the relation f ix = U/4 is valid, whence

F , = (nfi/4)[Q(-1) - Q(+1)1. (29)

When the mean free path is small compared with the distance over which Q changes appreciably, the quantities Q( - 1) and Q( + I ) in equation (29) may be expanded in a Taylor series about x = 0, and terms of order higher than the first may be neglected. The result is

F , = - (nd/2>(dQ/dx),. (30)

The final expression for the flux of Q in terms of the gradient of its average value is obtained by substituting equation (27) into equation (30), omitting the subscript 0 (since the choice of the position x = 0 was arbitrary)

640 Transport Properties

and generalizing to the case in which dQ/ay # 0 and aQ/az # 0 (by assuming that the medium is isotropic), namely,

(31) F = -(6/2,,h 0) v,Q. From equation (31) it is seen that the constant of proportionality between F and V, Q is independent of the number density and inversely proportional to the collision cross section; it is independent of pressure and its temperature dependence is -J'Tjo - T", where 3 I a I I, usually. Equation (31) is employed in the next two sections.

E.4. VISCOSITY

E.4.1. Coefficient of viscosity

Let u0,, be the average velocity of the molecules in the y direction, and let us investigate the transport of momentum in the y direction across planes normal to the x direction. If m is the mass of a molecule, the appro- priate value of Q is Q = muy, and F becomes the shear stress in the y direction on a surface with normal in the x direction [see Section D.21. In this case F is usually denoted by z,,, and equation (31) reduces to

zXy = -(m6/2& o) dv,,,/dx. (32) The coefficient of viscosity p may be defined conventionally as the

constant of proportionality between z X y and -duo, J d x in this geometry; that is,

zxy = - p dv,,,/dx, (33)

p = mii/2JZo. (34)

whence it follows that

Equation (34) implies that pis independent of pressure and that the tempera- ture dependence of p is p - f i / o - T", where 3 I a I 1.

The accurate kinetic theory for a one-component system yields [ IS]

p = 5k0T/8i2f2', (35)

which exhibits essentially the same pressure and temperature dependence predicted above. For hard-sphere molecules, equation (35) is found to reduce to

p = ( 5 7 ~ / 1 6 ) m 6 / 2 ~ 0, (36)

with which the approximate result in equation (34) fortuitously agrees within 2 %. Tables giving viscosity coefficients may be found in [ S ] , [6], and [18], for example.

E.4. Viscosity 64 1

For binary and multicomponent mixtures of gases the viscosity coefficient depends on the concentrations, and the results of the accurate kinetic theory are quite complicated. In terms of the viscosities of the pure of the pure components at the same pressure and temperature, pi, a useful empirical formula is [5J, [9J.

which (except for a factor of 2 in place of 1.385) has some theoretical justi- fication [ S ] . For most of the combustion applications discussed in this book, it is seldom practical to account for the concentration dependence of p.

The kinematic viscosity, p/p, has the same units as binary diffusion coefficients and the same type of pressure and temperature dependence, p/p - T"/p, with $ 5 crs 2. At atmospheric pressure, its values typically range from 10- cm2/s to 1 cm2/s for gases in combustion problems. The increase in viscosity with increasing temperature can be an important effect in the fluid dynamics of combustion.

E.4.2. The pressure tensor

The appropriate generalization of equation (33) to arbitrary geom- etries in multicomponent mixtures can be shown from kinetic theory [5J or from a reasonable continuum treatment to be

where y is a second viscosity coefficient, the superscript T denotes the trans- pose of the tensor, and the (symmetric) shear tensor T is related to the pressure tensor P [equation (D-21)] and to the hydrostatic pressure p by the equation

P = p U + T.

The shear stress T , ~ appearing in equation (33) is the xy component of T. Equation (39) clearly reduces to P = pU in equilibrium as is necessary (see Section D-2).

In quoting equations (38) and (39) in Chapter 1, the subscript 0 is omitted from vo and the second viscosity coefficient is replaced according to the expression

(39)

y = $ - K , (40)

where K 2 0 is the coefficient of bulk viscosity. For monatomic gas mixtures, kinetic theory shows that v = $ p ( K = 0). For polyatomic gases, relaxation effects between the translational motion and the various internal degrees

642 Transport Properties

of freedom may in some cases lead to a positive value of li," which is essentially proportional to relaxation time for small relaxation times and is zero for large relaxation times [S]. Few theoretical numerical calculations or reliable experimental measurements of IC yet exist, and bulk viscosity usually is negligible in combustion processes.

E.5. HEAT FLUX

E.5.1. Thermal conductivity

In order to treat thermal conduction in one-component systems, we may let Q = U i = U , the total energy of the molecule defined in equation (D-23). Then F becomes the energy per second transported across a surface of unit area; that is, F is the heat-flux vector q [see equations (D-28) and (D-29) with qR neglected]. Hence equation (31) becomes

q = - ( v / 2 4 OIV, B = -(mc,v/2JZ a>v, T, (41)

where c, is the specific heat at constant volume (that is, du/dT = me,).

constant of proportionality between q and - V, T ; namely, The thermal conductivity A of a one-component gas is defined as the

q = - AV, T. (42)

Equations (41) and (42) then show that

A = mc,v/2JZ = pc , , (43)

where use has been made of equation (34) in the last equality. Since c, is independent of pressure and usually depends only weakly on temperature, equation (43) implies that A is independent of pressure and that, roughly,

The accurate kinetic theory for a pure monatomic ideal gas [c, =

(44)

which agrees with equation (43) in the pressure and temperature dependence of A but differs in magnitude by a factor of 2.5, thus emphasizing the approxi- mate nature of the treatment in Section E.3. For pure polyatomic gases a rigorous kinetic theory expression for 1 has not yet provided useful numerical results [24]. The Eucken formula [25], based on the physical reasoning

I. - T",: I I 1.

$(kO/m)] gives [S]

A = % ( k O T c , / Q y ' ) = $/Lev,

* The pressure p appearing in equation (39) is not given by equation (D-22) when K # 0, since equations (38) and (39) imply (tr P)/3 = p - K ( V , . v,,). reflecting the fact that when translational-internal relaxation processes are of importance, the translational tem- perature is not the appropriate temperature to associate with the hydrostatic pressure [3].

E.5. Heat Flux 643

outlined below, is, therefore still the simplest way to compute thermal conductivities of polyatomic gases, although good experimental data are preferable when available.

Since the translational energy is proportional to the square of the velocity and the energy associated with internal degrees of freedom is ap- proximately independent of the velocity of the molecule (see Section D.2), the translational and internal energies should be transported differently, leading to different constants of proportionality between I I and pc, for each contribution. Equation (44) should be valid for the translational contribution, but the internal part will be transported more like momentum, whence 1 = pc, may be approximately true for the internal contribution.* Adding the two contributions therefore yields

1 = 2 2 (3ii") 2 m p + (c, - ;;) p,

which may be written as

by utilizing the fact that the specific heat at constant pressure is c p = c, + ko/m and the definition of the heat-capacity ratio y = cp/c , .

For binary and multicomponent mixtures, the thermal conductivity depends on the concentrations as well as on temperature, and the formulas of the accurate kinetic theory are quite complicated [ S ] . Empirical ex- pressions for I I are therefore more useful for both binary [9] and ternary [6], [26] mixtures, although few data exist for ternary mixtures. Tabulations of available experimental and theoretical results for thermal conductivities may be found in [ S ] , [6], [13], and [18]-[21], for example. The thermal diffusivity, defined as Ap/cp, often arises in combustion problems; its pressure and temperature dependences in gases are Alpc, - T"/p (3 I cx I 2), and its typical values in combustion lie between lo- ' cm'/s and 1 cmz/s at at- mospheric pressure.

E.5.2. The heat-flux vector

While equation (42) is valid for one-component systems without radiant transport, for binary and multicomponent mixtures there are other effects besides thermal conduction that contribute to the heat flux q.

* Although this might not be expected to be exactly true because the momentum is proportional to the first, not zeroth, power of the velocity, the derivation in Section E.4.1 fundamentally involved the component of momentum in the y direction and the component of velocity in the x direction; these are independent in a first approximation, and therefore the analogy is reasonable.

644 Transport Properties

When the average velocity of any component i differs from the mass- average velocity of the mixture, then n i V i molecules of type i per unit area per second flow across a surface moving with the mass-average velocity of tile gas mixture. If the enthalpy associated with an i molecule is denoted by H i = Ui + koT, these molecules will carry across the surface an average enthalpy per unit area per second equal to

RiniVi, where Ri = Di + koT

and 57, is given by equation (D-24). Denoting the average enthalpy per unit mass of species i by hi = H i / m i , we find from equations (D-5), (D-6), and (D-7) that H i n i V i = p h i y V i , whence the total enthalpy (of all species) per unit area per second flowing relative to the mass-average motion of the mixture is

N

i = 1

This term constitutes an additional contribution to q in binary and multi- component systems.

Onsager's reciprocal relations of irreversible thermodynamics [27-301 imply that if temperature gradients give rise to diffusion velocities (thermal diffusion), then concentration gradients must produce a heat flux. This reciprocal cross-transport process, known as the Dufour effect, provides another additive contribution to q. It is conventional to express the concen- tration gradients in terms of differences in diffusion velocities by using the diffusion equation, after which it is found that the Dufour heat flux is [5].

N N ROT ( X j D T , i / M ( D i j ) ( v i - Yj ) .

i = 1 j = 1

In most cases the Dufour effect is so small that it apparently often is negligible even when thermal diffusion is not negligible. Although it is omitted in the applications, this term is retained in the general equations for completeness.

Inserting this radiation flux and the terms discussed in the preceding paragraphs into equation (42) yields

N q = -AV,T + p 2 h i x V i +ROT

i = 1 j = 1

(46) as the complete expression for the heat-flux vector. The bars on Ti and the subscript x on V , are omitted when equation (46) is quoted in Chapter 1.

Calculation of qR necessitates consideration of radiation transport [31]-[33]. The spectral intensity IJx, a, t ) is defined as the radiant energy per unit area per second, traveling in a direction defined by the unit vector 0, per unit solid angle about that direction, per unit frequency range about

E.5. Heat Flux 645

the frequency v, at position x and time t. The relationship between q, and I, is

where dv and dR are elements of frequency and of solid angle, respectively, and the integration in R space extends over the entire solid angle 4rc. State- ment of an equation for I, necessitates introduction of absorption and scat- tering coefficients. The mass-based spectral absorption coefficient K , is defined such that p ~ , is the fractional decrease in radiant intensity per unit length of beam travel, due to absorption. The mass-based spectral scattering function 0, is defined such that the increase of I , per unit length of ray travel due to scattering of radiation from rays traveling in other directions, defined by unit vectors R’, is p @ ovlk dR’, where 1:. = IJx, R‘, t).* With these definitions, the equation of radiation transport may be written as

R * V,I, = ~ K , ( B , - I , ) + p ~ “ ( 1 ; - I , ) dR’ (48) il where the Planck function is

B, = (2hv3/c’)/(e”’koT - I), (49)

in which h and c are Planck’s constant and the velocity of light, respectively. Equation (48) states that, in the direction of ray travel, I, changes because of emission ( ~ K ~ B , ) , absorption ( - p ~ , l , , ) , and scattering (the last term). The scattering term exhibits both a gain from rays traveling in other direc- tions and a loss through scattering of the ray under consideration. Since equation (48) is an integrodifferential equation when scattering is important, correct inclusion of q, in equation (46) through use of equation (47) some- times may prevent combustion problems from being formulated in terms of partial differential equations.

In view of the complexity associated with equation (48), approximate methods are needed for applications. References [6] and [33]-[38] may be consulted for these approximations. While scattering may be important in combustion situations involving large numbers of small condensed-phase particles, often the effects of scattering may be approximated as additional contributions to emission and absorption, thereby eliminating the integral term. Two classical limits in radiation-transport theory are those of optically thick and optically thin media; the former limit seldom is applicable in combustion, while the latter often is. In the optically thin limit, gas-phase

* Note that ( ~ K J ’ and (puJ represent characteristic lengths for absorption and for scattering, respectively.

646 Transport Properties

absorption is small; if this and scattering are neglected in equation (48), then equation (47) may easily be shown to imply that

V, - q R = 4rcp lom K,B, dv = 4oT4/1,, (50)

where o = 2n5k04/15c2h3 is the Stefan-Boltzmann constant and 1, is the Planck-mean absorption length, which is defined by equation (50). Integra- tion of equation (50) shows that at the surface of a slab of thickness L, having uniform properties inside and no radiant flux incident on the back face, the component of qR normal to the surface of the slab is

q, = 4c~T~L/1, toT4, (51) where t = 4L/1, is the engineering emissivity of the slab. More generally, it is found that if absorption is included as well in this geometry, then

= 1 - ,-4L/b.

Data in references quoted above may be employed in equation (50) or (51) to estimate effects of radiation in combustion problems.

Often equation (51) is used to obtain radiant fluxes emitted from re- action regions; accompanied by geometrical considerations, it then provides estimates of radiant fluxes incident on solid or liquid surfaces near reaction regions. In many problems, the only significant effects of radiation are those concerning radiant transfer to and from solid surfaces. The normal compon- ent of radiant flux emitted from a solid surface is given by equation (51), with t representing the surface emissivity. Absorption lengths often are small for solids, so that their surfaces may be characterized by an absorptivity (which may be shown to equal t) and a reflectivity, 1 - E , having properties such that the fraction t of the incident energy flux is absorbed and the fraction 1 - t reflected, transmission being negligible. Improved descriptions of interaction of radiation with surfaces need to consider transmission, spectral variations and influences of incident angle (see references quoted above). There are a number of combustion problems in which it is important to employ equation (5 1) to calculate radiant losses from solid surfaces while all other influences of radiation remain negligible.

(52)

E.6. DIMENSIONLESS RATIOS OF TRANSPORT COEFFICIENTS

In flow problems, the conservation equations often can be written in forms involving dimensionless ratios of various transport coefficients. These ratios are defined in the present section. They refer to molecular transport, not to corresponding quantities that often have been defined for turbulent flows.

References 641

The Prandtl number, which is a rough measure of the relative impor- tance of momentum transfer and heat transfer, is defined as

Pr = pcp/A,

Pr = 4y/(9y - 5) which is given by

(53)

(54) for a pure polyatomic gas according to equation (45). From equation (54) we see that Pr, which is approximately 0.74 for air (y = 1.4), varies from 5 for monatomic gases (y = 5) to unity as y -+ 1. In some systems, the Eucken formula is not accurate and Pr can differ considerably from unity (for example, it is large compared with unity for liquids).

The Schmidt number, a rough measure of the relative importance of momentum transfer and mass transfer, is defined by

sc = PiPD12 (55) in a binary mixture. In multicomponent systems, Schmidt numbers may be defined for each pair of species. As in the case of the Prandtl number, Sc often is somewhat less than unity for gases; for liquids typically Sc % Pr.

The Lewis number (or Lewis-Semenov number) is a measure of the ratio of the energy transported by conduction to that transported by diffu- sion ; namely,

Le = A/pD,,c, (56) in a binary mixture. As with the Schmidt number, Lewis numbers may be defined for each pair of species in multicomponent mixtures. From equations (53), (55) and (56) it is apparent that

Le = Sc/Pr. (57) In many gases, Le is very nearly unity; it is often slightly less than unity in combustible gas mixtures. The approximation Le = 1 is frequently helpful in theoretical combustion analyses. However, small departures of Le from unity often can produce new phenomena in combustion. Therefore, a number of studies have been made of influences of the value of Le.

There are many other dimensionless groups that arise in applications involving transport processes [6], [39], [40]. Such quantities, a number of which are of practical importance in combustion [41], are introduced in the main text only to the extent that they are needed in the presentation.

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648 Transport Properties

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